Properties of the Global Solutions of a Two-Dimensional Incompressible Dissipative Quasi-Geostrophic Equation

Abstract

We consider a 2D quasi-geostrophic equation. We prove the existence of global smooth solution and weak solution to the Cauchy problem of this equation by using energy estimate. We also establish the elementary decay estimates and the primary decay estimate with sharp rate, the exact limits for all order derivatives of the global weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will consider our case for the initial function and the external force. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval’s identity, iteration technique, Lebesgue’s dominated convergence theorem, Cauchy-Schwartz’s inequality, etc) existing results (the existence of global weak solutions, the existence of local smooth solution on (T,∞).